Cowlitz River

Lake Scanewa

Estimating Fish Collection Efficiency

Estimating Fish Collection Efficiency

Estimating Fish Collection Efficiency

Estimating Fish Collection Efficiency

What about BTSPAS¹?

• Typically stratified by week
• Includes an explicit movement/delay component
• Model for probability of capture includes (independent) random effects
• Model for arrival rate depends on a temporal smoother
• Fit using JAGS²

Probability of Capture

Travel Time

• \(N_r\) fish are released on day \(r\)
• Travel time is modeled as a function \(F_{\theta}\)
• Proportion released on day \(r\) arriving on day \(t\) is

\[\theta_{rt} = \begin{cases} 0 & t \le r\\ F_{\theta}(t - r) - F_{\theta}(t - r - 1) & r < t, t - r \le T\\ 0 & i - t > T \end{cases}\]

• Assume no arrivals after day \(T\)

Probability of Capture

Proportion arriving each day

Probability of Capture

Probability of collection

• Given arrival, how likely is an individual to be collected?
• Modeled as \(p_t\), probability of collection on day \(t\)
• Flexible specification, allows covariates \(\boldsymbol{x}_t\)
• May use mgcv smoothers, including for time

\[p_{t} = \operatorname{logit}^{-1}(\phi_t) = f_{\phi}(\boldsymbol{x}_{t})\]

Probability of Capture

Observation Likelihood

For

• \(n_{rt}\) marked fish recaptured from release \(r\) on day \(t\) after release
• \(N_{r}\) total number of fish released on day \(r\)
• \(\theta_{rt}\) proportion of fish arriving on day \(t\) from release \(r\)
• \(p_t\) probability of capture on day \(t\)

\[[n_{r1},\ldots,n_{rT}, N_{r} - \sum_{t} n_{rt}] \sim\] \[\operatorname{Multinomial}(\theta_{r1} p_{r+1}, \ldots, \theta_{rT} p_{r+T}, 1 - \sum_{t} \theta_{rt} p_{t}\]

Arrival Process

• Log-arrival rate \(\lambda_{t}\) is modeled as a function of covariates \(\boldsymbol{z}_t\),

\[\log(\lambda_{t}) = f_{\lambda}(\boldsymbol{z}_{t}),\]

• \(U_{t}\) unmarked fish arrive on day \(t\) at rate \(\lambda_{t}\), so that

\[U_{t} \sim \operatorname{Poisson}(\lambda_{t})\]

• \(u_{t}\) individuals are collected with probability \(p_{t}\), so

\[u_t \sim \operatorname{Poisson}(\lambda_{t} p_t)\]

Collection efficiency

• Simulate values of \(U_t\) from a truncated Poisson distribution

\[U_t^{*} \sim \operatorname{Poisson}(\lambda_t) \quad \text{s.t.}\quad U_t \ge u_t\]

• Then estimated Fish Collection Efficiency is

\[\widehat{\textrm{FCE}} = \frac{\sum_t u_t}{\sum_t U_t^*}\]

Priors

• Covariates are scaled to have mean zero, standard deviation one
• Each coefficient has a \(Normal(0, \sigma_{i}^{2})\) prior
• \(\sigma_{i}\) is not currently estimated, including for smooths (where it acts as the smoothness penalty)

Model fitting

• Model is fit using the No-U-Turn Sampler in Stan

Cowlitz River Steelhead

Data

• 2020 to 2022
• Six releases each year
• 100 individuals per release¹
• No recaptures after 33 days

Cowlitz River Steelhead

Covariates

• Year intercept
• Two-way smooth with temperature and discharge
• Gaussian process smooth over the trapping season

Results

Partial effect of temperature and discharge

Results

Partial effect of temperature and discharge

Results

Estimated probability of capture

Results

Arrival rate estimate

Results

Run size estimate

Results

Fish Collection Efficiency

Can we do better?

• Assume temperature and discharge effects are shared among species
• Include separate intercepts and temporal smooths for Chinook and coho
• Adds 4 Chinook releases in 2020, 6 in 2021, and 7 in 2022
• Adds 11 coho release in 2020, 10 in 2021, and 9 in 2022

Results

Partial effect of temperature and discharge

Results

Estimated probability of capture

Results

Arrival rate estimate

Results

Run size estimate

Results

Fish Collection Efficiency

Future work

Other applications

• Other dam systems
• Smolt trap data

Improvements

• Allow travel time to vary over each season
• Account for more variation in arrival process
• Estimate smoothness penalty parameters
• Allow for batch-marked fish
• Allow for multiple recapture opportunities
• Create an R package for general use